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Product of reflection matrices without eigenvalue 1

  1. Jul 17, 2010 #1
    I'm wondering if anybody could suggest some techniques that might be brought to bear on the following problem:

    Suppose a finite sequence [tex]M_1,M_2,\dots,M_k[/tex] of [tex]4\times 4[/tex] orthogonal reflection matrices is given. I'm interested in determining conditions on these matrices that will guarantee that the product of any ordered sub-collection having an even and nonzero number of elements does not have eigenvalue 1.

    This is really the same as asking whether the result will be a simple or "double" rotation. If it's a simple rotation, the product will have an axis-plane that is fixed point-by-point and so will have eigenvalue 1. If it's a "double" rotation (two independent rotations in orthogonal 2-planes), then it will not have eigenvalue 1.

    Or better yet: does anybody have any ideas for generating a list of matrices satisfying the specified property (besides just randomly generating matrices and testing all generated collections)?
     
  2. jcsd
  3. Jul 17, 2010 #2
    Actually, I now don't think any such lists of 4x4 matrices exist.

    Claim: Given at least three reflections of [tex]R^4[/tex] in distinct hyperplanes through the origin, the composition of some two of them is a simple rotation fixing a two-dimensional plane through the origin(its axis).

    Proof(?): Maybe there's a proof using matrices directly, but my method uses the following connection with quaternions: the reflection of [tex]R^4[/tex] in the homogenous hyperplane orthogonal to the unit quaternion [tex]u[/tex] is the map [tex]q \mapsto -qvq[/tex] where [tex]v[/tex] is the conjugate of [tex]u[/tex]. (I can't get the \overline command in LaTeX to work here for some reason, so I will write the conjugate of [tex]u_i[/tex] as [tex]v_i[/tex] from here on out.) Another nontrivial observation, which follows from the previous one, is that the orthogonal rotations of [tex]R^4[/tex] are precisely the quaternion maps [tex]q \mapsto aqb[/tex] for unit quaternions [tex]a,b[/tex].

    Suppose three unit quaternions [tex]u_1,u_2,u_3[/tex] are given. The previous two observations imply [snip] that [tex]u_1 = u_2 = (v_1)^{-1}[/tex], and that's the end of the proof.

    Now, my vague understanding is that in higher dimensions the quaternions are replaced by the so-called Clifford algebras. Most sources I've looked at for Clifford algebras are a little too advanced for me. Does anybody know of a good, easy introduction to Clifford algebras?
     
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